Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: P1.92 - Parallel Block Adaptive Mesh Re?nement For Multiphase Flows
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00477
 Material Information
Title: P1.92 - Parallel Block Adaptive Mesh Re?nement For Multiphase Flows Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Zuzio, D.
Estivalezes, J.L.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: DNS
level set method
ghost fluid
AMR
multigrid
 Notes
Abstract: We present a two dimension parallelized adaptive algorithm for incompressible two phase flows. We use a coupled Level-Set and Ghost-Fluid method to track the interface between phases; the resolution of the Navier Stokes equations is performed by a two steps projection method. We make use of the PARAMESH package to implement the adaptive mesh and to get an efficient parallelization of our code. PARAMESH manages a block-based type AMR: the computational domain is recursively split into smaller subdomains where needed, until the desired resolution is reached. Special attention is paid to the elliptic solver for the pressure equation: we have developed a BiCGStab algorithm allowing the resolution of high density ratio flows over non uniform meshes; in order to speed-up the convergence we make use of a multigrid preconditioner based on a Fast Adaptive Composite algorithm. We refine the mesh on the interface to the maximum allowed level, leaving the rest of the domain to be less resolved: given the difficulty of solving the strong discontinuity lying on the interface, we want to show that a locally refined mesh can be as effective as the corresponding uniform fine mesh. Some validation test cases are proposed, such as the dynamics of static and rising bubble, and a Rayleigh-Taylor type instability.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00477
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: P192-Zuzio-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Parallel block adaptive mesh refinement for multiphase flows


D. Zuzio* and J-L. Estivalezes*


Department of Models For Aerodynamics and Energy, ONERA Centre de Toulouse
2, av. Edouard Belin BP 4025 31055 Toulouse Cedex 4, France
davide.zuzio ~onera.fr and Jean-Lue.Estivalezes ~onera.fr

Keywords: DNS, Level Set, Ghost Fluid, AMR, Multigrid




Abstract

We present a two dimension parallelized adaptive algorithm for incompressible two phase flows. We use a coupled
Level-Set and Ghost-Fluid method to track the interface between phases; the resolution of the Navier Stokes equations
is performed by a two steps projection method. We make use of the PARAMESH package to implement the adaptive
mesh and to get an efficient parallelization of our code. PARAMESH manages a block-based type AMR: the
computational domain is recursively split into smaller subdomains where needed, until the desired resolution is
reached. Special attention is paid to the elliptic solver for the pressure equation: we have developed a BiCGStab
algorithm allowing the resolution of high density ratio flows over non uniform meshes; in order to speed-up the
convergence we make use of a multigrid preconditioner based on a Fast Adaptive Composite algorithm. We refine
the mesh on the interface to the maximum allowed level, leaving the rest of the domain to be less resolved: given the
difficulty of solving the strong discontinuity lying on the interface, we want to show that a locally refined mesh can be
as effective as the corresponding uniform fine mesh. Some validation test cases are proposed, such as the dynamics of
static and rising bubble, and a Rayleigh-Taylor type instability.


Introduction


In the vast topic of flows with interfaces, the study
of liquid-gas interactions have a fundamental impor-
tance, especially in combustion problems. The inter-
facial instabilities, like the Kelvin-Helmholtz, produce
the droplets or sprays that afterwards participate to com-
bustion. In particular where experiments are too diffi-
cult or expensive to perform, the numerical simulation
becomes a powerful tool for predicting these physical
phenomena. The direct numerical simulation gives us a
"model free" approach to the Navier Stokes equations, at
the price, however, of a higher computational resources
demand. The strategy developed to improve the feasi-
bility of extended domain DNS are the parallel computa-
tion and the adaptative mesh. The PARAMESH package
(Peter MacNeice and Packer (2000)) is a set of libraries
developed to satisfy both approach: it is able to create an
adaptive mesh block type data structure, and to aggres-
sively distribute the computation among the processor
in order to achieve load balance. In a two fluid config-
uration the adaptive mesh is naturally advisable, as if
the precision of the interface discretization depends on
the resolution of the cells, it is often necessary to extend


the computational domain well beyond the interfacial re-
gion of the flow, thus involving a lot of computational
effort spent on less interesting zones. Berger (1982)
and co-workers have developed the use of a hierarchy
of logically Cartesian grids and sub-grids to cover the
computational domain. They allow for the sub-grids to
overlap, have arbitrary shapes and be merged with other
sub-grids at the same refinement level whenever appro-
priate. This is a flexible and memory-efficient strat-
egy, but the resulting code is complex and has proven
to be very difficult to parallelize. In a simplified variant,
the domain is divided in blocks which can be bisected
in each directions when needed (DeZeeuw and Powell
(1992)). This approach, followed by the PARAMESH
developers, intuitively produces meshes with lower re-
finement efficiency, but the dedicated tree memory stor-
age is lighter, and the cache managing as well as the
parallelization are easier. The resolution of an elliptic
equation for the pressure is an added difficulty for the
incompressible solvers, in particular when coupled with
the complex AMR mesh. The most attractive algorithms
are the Krilor subspace family methods, often coupled
with a preconditioner (Mark Sussman (1998), Teigland
and Eliassen (2001), Michael Oevermann (2000)). In







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


this work a preconditioned BiCG-Stab works in con-
junction with an AMR adapted multigrid. The adap-
tive mesh represents here an improvement over the al-
gorithm of Couderc and Estivalezes (2003); it does not
involve radical changes of the original code, and retains
the Cartesian structured mesh discretization.


Governing equations

Two fluids are considered; in each of them the density is
given and constant, and the flow is given by the incom-
pressible Navier-Stokes equations:


Numerical model

The variables are located into a staggered mesh, with
pressure and Level Set in the cell center and the veloc-
ity components on the cell faces. The Level Set linear
equation (3) spatial derivatives are discretized by a fi-
nite volume method and a fifth order Weno; the tempo-
ral derivative is integrated by a third order Runge Kutta.
The momentum equation (2) is solved by a second or-
der projection method: it works decoupling velocity and
pression by using first an explicit discretization for the
effects of convection in a predictor step, and then enforc-
ing the compliance with a velocity divergence constraint
in a subsequent projection step:

*resolution of the momentum equation without pres-
sure term


J 9 8x By


V -u = 0


~u
+ u- )u


- (V T +t f)


With the following hypothesis:

T = -pl+ D

D = p Vu + (Vu) )

where u [u, v]T represents the velocity vector field,
p the hydrodynamic pressure and p the density, p the
dynamic viscosity of the fluid, f represents the external
forces like gravity. The fluids are separated by a sharp
interface tracked via level set approach. It is implicitly
given by the zero level of a function (x, t), where 4
is defined as the signed distance to the interface, so that
any quantity a is


a (x, t) = Q(A) (x, t) if (x, t) > 0
al") (x, t) elsewhere

The evolution of the level set follows the linear advec-
tion equation

+ u V = (3)

so that it is passively advected by the velocity field. The
jump conditions imposed on the interface are determined
by capillarity and viscosity:


p 8 u"


8 u"


v?~~j 1 =Ui~+ At" i


u"


+v"


1
2


P d2V Z


d **
By"


*resolution of a Poisson equation, given the diver-
gence of the predicted velocities as right hand side


8J 1? 8p 8 18p
8x : p x y p y


} v


*correction of the intermediate velocities by the
pressure gradient


atn i
n+] *
U ~
i h~3 ,~3

n 1 atn
'U. 'U
"+1;3 i+ij+4


-


a/c (4)

0 (5)


[u]0


Here the notation [-] (-), (-)A represents the jump
across the interface, n the normal vector pointing from
A to B, t the tangential one. a is the surface tension, k
the interface curvature.


The viscous terms are treated by a totally explicit sec-
ond order centered scheme:


[p,] -n [p VBu + (Vu!)t n

t [I Vru + (Vu)~ t







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


" 2 '3 % t13t~a~j
a22


d2Un d2Un)i ,3
dz2 dy2 1





d2VndZ2 d2Vndy2


12 r
U
i 2~3 1


,j + Uni t~jl


ay2
n 2vn vn ~ 1
ij+ 2 %~3 2
a22
vn 'U
a+ Lj+l 2Vn 2 n
2 il~j
ay2


Figure 2: Representation of a 4x4 block with 2 guard-
cells in each direction.


than their parents. PARAMESH keeps also track of
physical boundaries on which particular boundary con-
ditions are to be enforced, ensuring that child blocks in-
herit this information when appropriate. The communi-
cation between coarse and fine grids is assured by the
prolongation operation, that involves giving values at
the fine points overlapping coarser ones, and the recip-
rocal one, the restriction. The interpolation algorithm
can be implemented by the user, or a default Lagrangian
polynomial interpolation is provided. The discretization
stencils are completed by a user defined set of ghost
cells around each block; Dirichlet boundary conditions
are exchanged among the blocks every synchronization
point: this structure allows the single grid discretization
to be applied to each block without modifications.
Conservation of mass is a key issue and to insure this
care must be taken at the interface between coarse and
fine patches. The fluxes along this interface are calcu-
lated on both the coarse and fine patches; these fluxes
do not necessarily match because they are evaluated on
different grids, so that mass could be lost or gained. To
insure mass conservation the flux through a coarse cell
face is therefore corrected to equal the sum of the fluxes
through the corresponding fine cell faces, a procedure
called flux matching shown in figure 3.


Poisson SOlver

In order to solve the linear Poisson pressure equation
(7) within the AMR framework we have combined the
BiCG-Stab algorithm (VanDerVorst (1992)) with a Fast
Adaptive Composite (McCormick and Thomas (1986))
multigrid preconditioner. The BiCG-Stab is an evolu-
tion of Krilor the subspace family methods CG, CGS


In this section the extension of the numerical method to
an adaptive mesh is explained. In the original Berger's
algorithm a certain number of cells are tagged for re-
finement, then clustered to form new finer grids which
are superimposed to the coarser ones; they can over-
lap and merge, but they have to be properly nested, so
that the refinement jumps are always one level. This
method is referred to as "patch based". In an alter-
native formulation, known as "block based", the com-
putational domain is split recursively into smaller ones
where needed, until the finest grid is generated (fig-
ure 1). In the present work, the PARAMESH package


Figure 1: Representation of the two kind of AMR, left
block-based, right patch-based.

(Peter MacNeice and Packer (2000),Olson and MacNe-
ice (2005),Olson (2006)) has been chosen to be used
with the code. It is an open-source software package
of Fortran90 subroutines developed by Peter McNeice
and Kevin Olson. The package manages the creation
of a block-type AMR: builds and maintains the tree-
structure which tracks the spatial relationships between
blocks, distributes them among the available processors
and handles all inter-block and inter-processor commu-
nication. It distributes the blocks so that locality is max-
imized and communications minimized. Each block is
an uniform cartesian mesh with fixed user defined di-
mensions, shown in figure 2; typically 2D blocks are
4x4 or 8x8. The refinement ratio is fixed to r = 2,
that means refined cells are always two times smaller


Adaptive mesh refinement















I .







Figure 3: Representation of the 2D flux match-
ing procedure. In black are represented the mesh
point; in grey the interpolated guardcells points.
Facarse =Fine + F/in /2


and BiCG. It is able to work with unsymmetrical linear
systems and does not require the explicit computation
of the system matrix transpose. When the Laplacian is
discretized on the adaptive mesh, the standard discretiza-
tion stencil is applied to the interior points of each block,
given the updated interpolated values in the guardcells.
The global matrix is hence unsymmetrical due to the in-
terpolation coefficients.
The multigrid preconditioning is widely em-
ployed Coudere (2007),Sussman (2005),Vuik and
J.J.I.M vangan (2000) to speed up the convergence. A
natural way to implement a multigrid algorithm within
an AMR mesh is to utilize the coarse grid to perform
the basic two grid cycle. The AMR tree can be "inter-
preted" in two different ways: a multigrid level may
coincide with a refinement level, so that the relaxation is
performed each level on a fraction of the computational
domain, or the level may be built with all the grid points
until the "local" level refinement is reached. The latter
has been chosen for some reason. The first is for parallel
efficiency purpose: in the PARAMESH framework the
repartition of the blocks among the processors is done
for the leaf blocks first, because the most of the time
is spent here. So, if a relaxation sweep is performed
on a fixed level of refinement, some processors only
will likely be at work. Instead, if it is done on the
entire domain at each level, it can benefit from a more
efficiently parallelized relaxation The second reason
is that PARAMESH multigrid routines are meant to
work in a global sense, so that it would not be possible
to easily perform "sub-multigrid" relaxation over a few
blocks only (like Sussman (2005)), and the FAC is
meant to work in a more global sense.
As for the advection equation, the coarse-fine inter-
face problem arises. The Dirichlet boundary condition
exchange between blocks makes coarse and fine solution
not sufficiently linked, since the pressure gradient are
computed on two different meshes. This causes a dis-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


continuity in the first derivative which is O(Axcourse),
that acts like a singular charge proportional to the first
derivative mismatch, and corrupts the solution prevent-
mng the attainment of the fine grid precision or even
convergence. The solution proposed here is, as sug-
gested by Teigland and Eliassen (2001), a Dirichlet and
Neumann boundary conditions coupling. Across the in-
terface a preliminary Dirichlet conditions exchange be-
tween blocks is allowed in the two ways; then the coarse
grid ghost cells values are corrected to make the first
derivative match the fine grid value, so that the coarse
grid actually receives an imposed Neumann boundary
values.



Numerical results

Static bubble

The simulation of a round static bubble is the first in-
teresting test for the numerical method. Although very
simple in theory, it can give informations about the pre-
cision of the interface treatment, in particular of the sur-
face tension. A round bubble of a dense fluid is cen-
tred in a square domain, surrounded by a lighter one;
there are no initial velocities and no gravity. In this case
the pressure jump is imposed by the interface bending
and the surface tension product, as given by the Laplace
equation


pint pext = a/R, if in 2D
Pint Pext = 2a/R, if in 3D


Any error resulting from the discrete computation of the
level set bending leads to an erroneous value of the pres-
sure gradient between two adjacent cells: this non zero
pressure gradient then acts as a wrong correction for the
predicted velocity field. This the cause of the so called
spurious currents, non zero components of the velocity
that arise from the first iteration, and may in some cases
destroy the interface. A comparison between two dif-
ferent test cases is proposed: in the first mesh conver-
gence results from some uniform meshes are shown; in
the second one the same computation is performed with
an adaptive mesh localized on the interface. The param-
eters of the simulation are taken from F. Coudere (2006):
L 4 ,cm, R 1 cm, pint 1000 kg.m-3, pext=
1 kg.m-3, Pint 10-3 Pa.s, Pext 10-sPa.s,
a 0.1 N.m- Only one iteration of level set reini-
tialization is performed in order to reduce any artificial
displacement. In the first table the norm of he velocity is
calculated after one time step alone: here the predicted
velocities are equal to zero, so that the spurious currents







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Mesh
Uniform


Grid
16
32
64
128
256
16
32
64
128


lI .I
1.933e-06
1.910e-07
2.485e-08
3.277e-09
3.893e-10
1.933e-06
2.125e-07
3.259e-08
5.717e-09


AMR


2562 9.451e-10 2.60

Table 1: Spurious currents norm, first iteration. AMR
grid correspond to the finest level.


little more than in the uniform mesh; still it does not rise
dangerously, but it stabilizes as for the uniform mesh.
The convergence rates in table 2 are now near to 1.5.


Figure 4: Pressure field of a static round bubble after one
timestep captued with four grid levels


are determined by the corbure computation.





If the analytical bending is used as input, then the solu-
tion is always precise to the roundoff error. As shown in
table 1, the convergence rate, given by

In (errg,) -- In (err,)
In (2)

are, for the uniform mesh computation, close to 3:
this behaviour, justified by Oevermann and Berger
Michael Oevermann (2000), comes from the zero gradi-
ent solution of the pressure on both the sides that nullify
the effects of the interface position relative to the grid.
In the adaptive mesh test a slight decrease of the code
performance, mainly for the finer resolutions, is visible.
It has to be kept in mind that there is no lower limit to
the refinement level, so that the lower resolution are kept
far from the interface as the resolution increases. The ef-
fects of the coarser grids and the interpolation are visible
in the small spurious current increase. Still, the conver-
gence rate never drops under 2.5.
The growth of the spurious currents are here checked
for a prolonged time of simulation: in this case the code
tries to achieve an equilibrium for those non-physical ve-
locities, with the generation of non zero vorticity near
the interface. An initial steep growth followed is fol-
lowed by a stabilization in time. In the AMR compu-
tation the initial error grows, as for the first iteration, a


Mesh
Uniform






AMR


Grid I
162 1.885E-03
322 5.828E-04 1.69
642 1.778E-04 1.71
1282 3.879E-05 2.20
2562 1.060E-05 1.87
162 1.885E-03
322 5.676E-04 1.73
642 2.181E-04 1.38
1282 5.545E-05 1.98
2562 2.953E-05 0.91


Table 2: Spurious currents norm, t = 1
correspond to the finest level.


s. AMR grid


Rising bubble

In this section the dynamics of a rising round bubble are
presented. The bubble is now immersed into a heavier
fluid and subjected to gravity. There are no analytical
solutions to compare; however, extensive research work
has been done in creating benchmarks for this problem,
mainly from Mark Sussman (2002) and Hysing et al.
(2008). In this sense, a convergence test for some com-
puted quantities and a comparison with the Hysing's are
performed.
In the initial configuration there is a column of fluid 1
with no slip conditions imposed on the horizontal sur-
faces and slip conditions on the vertical ones. In the







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Time [s]


Figure 5: Evolution of the spurious currents for a long
time simulation, t = 1 .s


lower half of the [1x2] domain, centered at [0.5, 0.5] is
the circular bubble of radius [r = 0.25] composed of a
lighter fluid, all the velocities initialized to zero. As the
gravity applies its effect, the bubble starts rising, and at
the same time begins to change its shape. As in the Hys-
ing's benchmarks, computations are performed for two
configurations.
Two dimensionless numbers can describe the regime
of the bubble rise. Given L 2rn and LT, ~
these are the Reynolds and the Eiitviis number:


Figure 6: Initial condition for the rising bubble test


This is an interesting quantity to compute, because it
does not only measure how the interface tracking algo-
rithm behaves, but also it gives an idea of the quality of
the overall solution; this quantity checks also the reduc-
tion of precision of the AMR grid far from the interface,
and its influence on the interface itself. The last quantity
is the circularity, an index of how much the shape of the
bubble differs from the circle:


pb
Here the numerator is the perimeter or circumference
of a circle with diameter cl,, which has an area equal to
that of a bubble with perimeter Pb: it always starts from
the maximum value of one, an then decreases as the bub-
ble begin to be deformed. Some qualitative comparisons
between our results and Hysing's ones are presented in
figure 9; for the quantitative results, the results of the re-
fined mesh are compared with a reference computation
made with an uniform fine mesh of [256x1024]. The dif-
ference between the two values is calculated at the final
timestep: e rr = ahbs(qt=3 9,ef ,t=3).
The temporal evolutions of the bubble values follow
the benchmarks for both the configurations; however he
code could not track the circularity for the second test
case after t 1.5 .<, approximately when the bubble
forms the two sharp corners. In this location Hysing
could find, with a finite elements code, a small ligament
that subsequently gives birth to two small drops.
In the first simulation set the convergence towards the
reference solution is very good. The position of the cen-


Eo =


Re =


In the first case the surface tension should be enough
to hold the bubble together, and the final shape should be
of an ellipsoidal regime, ; the density and viscosity ra-
tio are equal to 10. The second test is more challenging,
as the density ratio reaches 1000 and the viscosity ratio
100; this bubble lies somewhere between the skirted and
dimpled ellipsoidal-cap regimes indicating that break up
can possibly occur (Clift et al. (1978)). The evolution
of the two bubbles is tracked for three time units, during
which three quantities are measured. The first is the cen-
troid of mass (.re, yc), of which, given the symmetry of
the problem, the y coordinate only is considered:





The second is the instantaneous rise velocity (u, v),
again y component only:


SI22 V cl#
foP cy


o "








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Test case
1


#1g #2 y Re
10 1 0.98 24.5 35


P1/# #/#
10 10


P1 #2
1000 100


2 1000 1


10 0.1 0.98 1.96 35 125 1000 100


Table 3: Physical parameters for the two test cases


Mesh value err
Ye 32 1.0894 4.03E-3
64 1.0863 9.39E-4


1.0856
1.0854
0.9313
0.9233
0.9206
0.9212
0.1986
0.1964
0.1955
0.5917


2.39E-4


1.01E-2
2.16E-3
5.17E-4


2.86E-3
6.98E-4
1.57E-4


Vcise


Table 4: Computed values for the rising bubble, test 1.
Mesh value refers to the finest level.


(a) Solution test 1


(b) Solution test 2


Figure 7: Solution after t = 3 .s of the two configurations
of bubble, four levels of mesh up to 128x256


Mesh value
Ye 32 1.1619
64 1.1525
128 1.1477
ref 1.1434
32 0.8630
64 0.8532
128 0.8443
ref 0.8389


err
1.86E-2
9.21E-3
4.42E-3


2.41E-2
1.43E-2
5.38E-3


(b) Hysing, Re = 35, Eo
125


(a) Hysing, Re 35, Eo =
10


Figure 8: Zoom over the bubble shape, Hysing


ter of mass is almost the same for all the meshes, and
grows almost linearly in time; the convergence rate is
quadratic for this value. The circularity computation is
more difficult: the lowest mesh computed value seems
quite far from the good solution, mainly in the last half
of the simulation; still there are similar values of q. The
rise velocity seems a little bit underestimated, for the
lesser mesh, in the region around the maximum; the
terminal velocity is however well estimated for all the
grids, and q keeps quadratic.
The second test case is clearly more difficult. Again
the center of mass rises linearly in time; however, we
have linear convergence rate for its final position. The


0.2105 5.61E-3
0.2125 3.59E-3


Vcise


0.2144
0.2161


1.66e-3


Table 5: Computed values for the rising bubble, test 2.
Mesh value refers to the finest level.



circularity tracking is lost for all the meshes after ap-
proximately 1.75 .s. The instantaneous mean rise veloc-
ity improves quite a lot with the finer meshes: with a
resolution of at least Ar: = 1/64 the second velocity
peak is found around t = 2 .s. For those values the con-
vergence rate is linear or sublinear.




































Time [s]
(b) Center of mass, test 2


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


r v 1/32
1/64

+ Hysing [1/320] -


1





~0.8 -




0.6-


r r1/32
v 1/64

+ Hysing [1/320]


Y,


.~.I


Time [s]
(a) Center of mass, test 1


0.8-




AA 1/32
r v1/64
1/128
0.6 + Hysing [1/320]


4,


Z +i.~


.- 1/128
+ Hysing [1/320]












1 2
Time [s]
(c)( l C I .. 1 I test 1


0.98 E


.
I,


. 0.96


Time [s]
(d)(l CI ...ni .. test 2


02 i h.*~*Ysys..r.~~;;;~~':;I


0.15 -




0.05



0.0

0o


0.15

n .


--- 1/32


+ Hysing [1/320]


Time [s]
(e) Rise velocity test 1


Time [s]
(f) Rise velocity test 2


Figure 9: Comparison of the temporal evolution of three quantities







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Acknowledgements

This research project has been supported by a Marie
Curie Early Stage Research Training Fellowship of the
European Communitys Sixth Framework Programme
under contract number MEST-CT-2005-020426.
The PARAMESH software used in this work was
developed at the NASA Goddard Space Flight Cen-
ter and Drexel University under NASA's HPCC and
ESTO/CT projects and under grant NNGO4GP79G from
the NASA/AISR project.

References

M. J. Berger. Adaptive mesh refinement for hyperbolic
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Rayleigh-Taylor instability

In this section the simulation of a Rayleigh-Taylor insta-
bility is presented. This phenomenon can be observed
by superimposing a heavy fluid over a lighter one; with-
out any perturbation they keep in equilibrium. An initial
sinusoidal perturbation of a given wavelength is subject
to the destabilizing effect of gravity and the stabilizing
effect of surface tension: if the destabilizing effect pre-
vails, then the perturbation grows, as the heavier fluid
moves down and deplaces the lighter one. The simu-
lation is performed in a L, 1 m, L, 4 m do-
main. The interface is placed horizontally in the mid-
dle of the domain. The computational parameters are:
ph = .22o kg.m ,pi 0.1694 kg.m ,PA = t
3.13E 3 kg.m- .sl ,o 0 N.m- The initial per-
turbation is y 0.05cos(kx) with k 2x7. Those pa-
rameters are the same used by Popinet Popinet (2000):
into his work a comparison is made between an eule-
rian VOF method and a lagrangian type interface track-
ing. In this simulation the interface sustains strong non
linear deformations, the initial perturbation grows into
a "mushroom" shape until 0.8 seconds. At this time the
extremities begin to form two thin ligaments that quickly
shrink to the grid cell size. With insufficient resolution
the ligaments are broken and, subsequently, two drops
are detached, so that the computed velocity field deviates
quickly from the good solution. The precise lagrangian
solution shows us that until t = 0.9 s the ligaments are
kept. We have performed the first computation with the
same base resolution, 64 x 256 with four levels of AMR.
We can see that at the last timestep the ligaments are al-
ready somewhat fragmented, even if the shape remains.
We have subsequently tried to improve the interface res-
olution by adding an AMR level, reaching a fine reso-
lution of 128 x 512: in this simulation the ligaments are
clearly maintained, showing us the improvement on of
the biphasic code by a local refinement over the inter-
face.



Conclusions


A second order parallel adaptive projection method for
incompressible two phase flows has been developed.
The ability to perform high density ratio computations
has been demonstrated, in particular the ability of the
elliptic solver to converge for all the test cases. In this
paper it is proven that a local interface refinement AMR
simulations can nearly achieve the fine grid precision in
all the tests, as well as maintaining good convergence
rate. An extension to three dimension will soon be done,
as the benefit of the AMR would be even more evident.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Figure 10: Results from the Rayeigh-Taylor instability computations, five levels of mesh, finest grid equivalent to
64x2.56. t = 0.9 s


Figure 11: Results from the Rayeigh-Taylor instability computations, four levels of mesh, finest grid equivalent to
128x512, t = 0.9 s







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Kevin Olson. Paramesh: A parallel adaptive grid tool.
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differential equations. European Congress on Compu-
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